Let \(\mathcal {O}^{\prime } \Subset \mathcal {O}\)  and \(M:=\Vert  \phi \Vert _{\mathcal {L}^\infty (\mathcal {O}^{\prime }, \mathbb {R})} <+\infty \) .
We split the proof in several steps. First we use Friedrichs' mollifier to regularize \(\chi _{E }\)  with a family of functions \((f_\alpha )_{\alpha >0}\)  which satisfies Theorem REF . Then we consider the level set of \((f_\alpha )_{\alpha >0}\)  which defines a family of functions \((\phi _\alpha )_{\alpha >0}\) . Finally we prove that \((\phi _\alpha )_{\alpha >0}\)  is the family of functions, up to subsequence, that we were looking for.
\(\mathbf {Step \, 1.}\)  Let \(\mathcal {E}:=\mathcal {E}_{\hat{\phi }}\)  the intrinsic subgraph of \(\hat{\phi }|_{\mathcal {O}^{\prime } }\)  defined as (). For every \(\alpha >0\)  we consider \(f_\alpha :\mathbb {G}\rightarrow \mathbb {R}\)  given by
\(\begin{aligned}f_\alpha (p):= (\rho _\alpha \ast \chi _{\mathcal {E}})(p)& =\int _\mathbb {G}\rho _\alpha (pq^{-1}) \chi _{\mathcal {E}}(q)\, d \mathcal {L}^{m+n}(q)\\& =\int _\mathbb {G}\rho _\alpha (q) \chi _{\mathcal {E}}(q^{-1} p)\, d \mathcal {L}^{m+n}(q)\end{aligned}\) 
where \(\rho _\alpha (p):=\alpha ^{m+2n}\rho (\delta _{1/\alpha }(p))\)  and \(\rho \in \infty _c (\mathcal {U}(0,1), \mathbb {R})\)  is a smooth mollifier with \(\rho (p^{-1})=\rho (-p)=\rho (p)\)  for every \(p\in \mathbb {G}\) .
By properties of convolution in \(\mathbb {G}\)  introduced in [1]}, we know that \(f_\alpha \in \infty _c (\mathbb {G}, \mathbb {R})\)  and spt\((f_\alpha )\subset \mathcal {U}(0,\alpha )\cdot \)  spt\((\chi _{\mathcal {E}})\)  for every \(\alpha >0\) . Precisely, for all \(\alpha >0\)
\(f_\alpha (p)\in [0,1] \quad \mbox{ for all } p\in \mathbb {G}\) 
and for all sufficiently small \(\alpha >0\)
\(f_\alpha (p)=1 \quad \mbox{ for all } p=(p_1,\dots , p_{m+n})\in \mathbb {G}\, \mbox{ such that } p_1 \le -2M\) 
Moreover by definition of \(\mathcal {E}\)  we know that \(\mathcal {E}\)  is an open set in \(\mathbb {G}\)  and
\(\mbox{ spt} (\chi _{\mathcal {E}}) = \mbox{clos} ( \mathcal {E}) \subset \lbrace a\cdot (t,0,\dots , 0 )\in \mbox{clos} (\hat{ \mathcal {O}^{\prime }} ) \cdot :\, t \le M \rbrace .\) 
As a consequence
\(\mbox{ spt} (f_\alpha ) \subset \overline{\mathcal {U}(0,\alpha )}\cdot \mbox{ spt} (\chi _{\mathcal {E}})\subset \lbrace a\cdot (t,0,\dots , 0 )\in \hat{ \mathcal {O}^{\prime }} \cdot :\, t < 2M \rbrace  \quad \mbox{ for } \alpha <M\) 
and
\(f_\alpha (p)=0 \quad \mbox{ for all } p=(p_1,\dots , p_{m+n})\in \mathbb {G}\, \mbox{ such that } p_1 \ge 2M\) 
i.e. \(f_\alpha \)  is constant far from the graph of \(\hat{\phi }\) , so \((\ref {conv})\)  is extended only in a neighborhood of the graphs itself.
\(\mathbf {Step \, 2.}\)  For every \(\alpha >0\)  and \(c\in (0,1)\) , let
\(S_\alpha := \left\lbrace  p=(p_1,\dots , p_{m+n})\in \hat{ \mathcal {O}^{\prime }} \cdot  : \, p_1 \in (-2M,2M),\, f_\alpha (p)=c \right\rbrace .\) 
We note that if
\(\mbox{rank} \nabla _\mathbb {G}f_\alpha (p)=1, \quad \mbox{for all } p\in S_\alpha \) 
then \(S_\alpha \)  is the level set of a map \(f_\alpha \in \infty (\mathbb {G}, \mathbb {R})\)  and so \(f_\alpha \in 1_\mathbb {G}(\mathbb {G}, \mathbb {R})\)  such that \(\mbox{rank} \nabla _\mathbb {G}f_\alpha \)  is maximum, i.e. \(S_\alpha \)  is \(\mathbb {G}\) -regular hypersurface. More precisely, we show that
\(X_1f_\alpha (p) <0 \quad \mbox{for all } p\in S_\alpha \) 
where \(X_1f_\alpha (p)\)  is the first component of horizontal gradient \(\nabla _\mathbb {G}f_\alpha \)  of \( f_\alpha \)  in \(p\) . Indeed let \(\mathcal {A} ^{\prime }:=\lbrace p= (p_1,\dots , p_{m+n})\in \hat{ \mathcal {O}^{\prime }} \cdot  :\, p_1\in (-3M,3M) \rbrace \)  and \(\varphi \in \infty _c(\mathcal {A} ^{\prime } , \mathbb {R})\) , then
\(\begin{aligned}\langle X_1f_\alpha , \varphi \rangle &=-\int _{ \mathcal {A} ^{\prime } } f_\alpha (q)X_1 \varphi (q) \, d\mathcal {L}^{m+n} (q)\\&=-\int _{ \overline{\mathcal {U}(0,\alpha )} } \rho _\alpha (p) \, d\mathcal {L}^{m+n} (p) \int _{ \mathcal {A} ^{\prime } } \chi _\mathcal {E}(p^{-1}q) X_1 \varphi (q)\, d\mathcal {L}^{m+n} (q)\\&=-\int _{ \overline{\mathcal {U}(0,\alpha )} } \rho _\alpha (p) \, d\mathcal {L}^{m+n} (p) \int _{\tau _{P^{-1}} (\mathcal {A} ^{\prime }) } \chi _\mathcal {E}(q) X_1 \varphi (pq)\, d\mathcal {L}^{m+n} (q)\\\end{aligned}\) 
With the notation \(\varphi _p(q):=\varphi (pq)\)  we have \(X_1(\varphi (pq))= X_1 \varphi _p (q)\)  because \(X_1\)  is left-invariant; moreover \(\varphi _p \in \infty _c (\tau _{p^{-1}} \mathcal {A} ^{\prime } , \mathbb {R})\)  and by integration by parts, we obtain
\(\begin{aligned}\int _{\tau _{P^{-1}} (\mathcal {A} ^{\prime }) } \chi _\mathcal {E}(q) X_1 \varphi _P(q)\, d\mathcal {L}^{m+n} (q) = - \int _{\tau _{p^{-1}} (\mathcal {A} ^{\prime }) } \nu ^{(1)}_{\mathcal {E}} (q) \varphi _P(q)\, d |\partial \mathcal {E}|_\mathbb {G}(q)\end{aligned}\) 
Since \(\mbox{spt}(\varphi _P) \Subset \tau _{P^{-1}} (\mathcal {A} ^{\prime })\)  and \(P\in \overline{\mathcal {U}(0,\alpha )}\)  if \(\alpha \)  is small enough, we can replace \(\tau _{p^{-1}} (\mathcal {A} ^{\prime })\)  by \(\mathcal {A} ^{\prime }\) . Thus by Fubini-Tonelli Theorem and a change of variable, we get
\(\begin{aligned}\langle X_1f_\alpha , \varphi \rangle &=\int _{\mathcal {A} ^{\prime } } \nu ^{(1)}_{\mathcal {E}} (q) \, d |\partial \mathcal {E}|_\mathbb {G}(q) \left( \int _{\mathbb {G}} \rho _\alpha (p) \varphi (pq) \, d\mathcal {L}^{m+n} (p) \right)\\\end{aligned}\) 
Then for all \(p\in \mathcal {A} ^{\prime }\)  and for all \(\alpha >0\)  small enough
\(\begin{aligned}X_1f_\alpha (p) & = \int _{ \mathcal {A} ^{\prime } }\rho _\alpha (p q^{-1}) \nu ^{(1)}_{\mathcal {E}} (q)\, d |\partial \mathcal {E}|_\mathbb {G}(q)\\&= \int _{\mathcal {U}(0,\alpha )\cdot P }\rho _\alpha (p q^{-1}) \nu ^{(1)}_{\mathcal {E}} (q)\, d |\partial \mathcal {E}|_\mathbb {G}(q)\end{aligned}\) 
In particular we immediately deduce from \((\ref {83cit})\)  the following assertion: For all couple \((\hat{ \mathcal {O}^{\prime }} ,\hat{ \mathcal {O}} _0^{\prime })\)  of open and bounded subsets of \(\mathbb {W}\)  with \(\hat{ \mathcal {O}^{\prime }} \Subset \hat{ \mathcal {O}} _0^{\prime }\)  there is \(\bar{\alpha }=\alpha (\hat{ \mathcal {O}} _0^{\prime })>0\)  such that for every \(0<\alpha <\bar{\alpha }\)
\(\int _{ \mathcal {A} }|\nabla _\mathbb {G}f_\alpha |\, d \mathcal {L}^{m+n} \le |\partial \mathcal {E}|_\mathbb {G}( \mathcal {A}_0 ).\) 
where \( \mathcal {A}:=\lbrace p= (p_1,\dots , p_{m+n})\in \hat{ \mathcal {O}^{\prime }} \cdot  :\, p_1\in (-2M,2M) \rbrace \)  and \( \mathcal {A}_0 :=\lbrace p= (p_1,\dots , p_{m+n})\in \hat{ \mathcal {O}}_0^{\prime } \cdot  :\, p_1\in (-2M,2M) \rbrace \) . Moreover since Lemma \(\ref {coro4.2}\)  we get
\(\nu ^{(1)}_{\mathcal {E}} \circ \Phi =-\frac{1}{\sqrt{1+|\nabla ^{\hat{\phi }} \hat{\phi }|_{ \mathbb {R}^{m-1}}^2}}\quad \mbox{in } \mathcal {O}^{\prime }\) 
and if we put
\(I_\alpha (p):= \int _{ \mathcal {U}(0,\alpha )\cdot p}\rho _\alpha (pq^{-1})\, d |\partial \mathcal {E}|_\mathbb {G}(q)\) 
then using \((\ref {83cit})\)  we have
\(X_1f_\alpha (p) \le -\frac{1}{\sqrt{1+\Vert \nabla ^{\hat{\phi }} \hat{\phi }\Vert ^2_{\mathcal {L}^\infty (\hat{ \mathcal {O}^{\prime }} , \mathbb {R}^{m-1})}}}I_\alpha (p) \quad \mbox{ for all } p \in \mathcal {A}^{\prime } .\) 
Moreover for every fixed \(c\in (0,1)\)  and for all \(p\in \mathcal {A}\)  such that \(f_\alpha (p)=c\) , then
\(\mathcal {L}^{m+n}((\mathcal {U}(0,\alpha )\cdot p) \cap \mathcal {E})>0 \quad \mbox{and} \quad \mathcal {L}^{m+n}((\mathcal {U}(0,\alpha )\cdot p) \cap \mathcal {E}^c )>0.\) 
Indeed, by contradiction, if we assume \( \mathcal {L}^{m+n}((\mathcal {U}(0,\alpha )\cdot p) \cap \mathcal {E})=0\) . Then because \(\mathcal {E}\)  is open, we can assume \((\mathcal {U}(0,\alpha )\cdot p) \cap \mathcal {E}= \emptyset \) . Hence by the definition of convolution, we have that \(f_\alpha (p)=0 \ne c\)  and then a contradiction. In the similar way it follows that \(f_\alpha (p)=1\)  if we suppose by contradiction \( \mathcal {L}^{m+n}((\mathcal {U}(0,\alpha )\cdot p) \cap \mathcal {E}^c)=0\) .
Now by \((\ref {cit88})\)  and by Theorem REF , we have that \(|\partial \mathcal {E}|_\mathbb {G}((\mathcal {U}(0,\alpha )\cdot p )>0\)  and \(I_\alpha (p )>0 \)  for \(p\in \mathcal {A} \)  with \(f_\alpha (p)=c\) . As a consequence using \((\ref {cit87})\) , \((\ref {cit85})\)  holds and \(S_\alpha \)  is a \(\mathbb {G}\) -regular hypersurface.
\(\mathbf {Step \, 3.}\)  Now we show that \((S_\alpha )_\alpha \)  implicitly defines a sequence \((\hat{\phi }_\alpha )_{\alpha }\)  with \(\hat{\phi }_\alpha =(\phi _\alpha ,0,\dots , 0)\)  and ,up to subsequence, it is the family of functions which we were looking for.
Fix \(\alpha \)  and \(c\in (0,1)\) . Because \(f_\alpha \)  is continuous map such that
\(\begin{aligned}f_\alpha ( i(a) \cdot (-2M,0,\dots , 0)) & =1>c \\f_\alpha (i(a)\cdot (2M,0,\dots , 0)) & = 0<c\end{aligned}\) 
for all \(a\in \mathcal {O}^{\prime } \) , there is \(t_a\in (-2M,2M)\)  such that \(f_\alpha ( i(a) \cdot (t_a,0 , \dots , 0))=c\) . In particular since
\(X_1f_\alpha ( i(a) \cdot (t,0,\dots , 0) )\le 0 \quad \forall \, t<t_a \, \qquad \mbox{ and } \qquad X_1f_\alpha ( i(a) \cdot (t_a,0,\dots , 0)) < 0\) 
we have that \(t_a\)  is unique and \( \lbrace  t\in (-2M,2M) \, :\, f_\alpha ( i(a) \cdot (t,0,\dots , 0))>c \rbrace =(-2M,t_a)\) .
In the similar way in Lemma 4.8 in [2]}, we define \(\hat{\phi }_\alpha (i(a)):=(\phi _\alpha (a),0,\dots , 0)\)  with \(\phi _\alpha (a):=t_a\)  and
\(\mathcal {E}_\alpha =\mathcal {E}_{\alpha , c}:= \lbrace  (i(a) \cdot (t,0,\dots , 0)) \in \hat{ \mathcal {O}} \cdot  :\, f_\alpha (i(a)\cdot (t,0,\dots , 0))>c\rbrace .\) 
Then \(\phi _\alpha :\mathcal {O}^{\prime } \rightarrow [-2M, 2M]\)  and
\(\mathcal {E}_\alpha \cap \mathcal {A}^{\prime \prime } =\mathcal {E}_{\hat{\phi }_\alpha } \cap \mathcal {A} ^{\prime \prime },\) 
where \( \mathcal {A}^{\prime \prime } :=\lbrace p\in \hat{ \mathcal {O}^{\prime }} \cdot  :\, p_1\in [-2M, 2M] \rbrace \) . Moreover recalling that \(c\in (0,1)\) , from \((\ref {cit76})\)  and \((\ref {cit79})\)  we get
\(S_\alpha = \mathrm {graph}\,(\hat{\phi }_\alpha ) =\partial \mathcal {E}_\alpha \cap ( \hat{ \mathcal {O}^{\prime }} \cdot \) 
and by Theorem REF  \(\hat{\phi }_\alpha \)  is uniformly intrinsic differentiable in \(\hat{ \mathcal {O}^{\prime }} \) .
Let \(\tilde{f} _\alpha :\mathbb {R}^{m+n} \rightarrow \mathbb {R}\)  defined by
\( \tilde{f}_\alpha (t,a ):=f_\alpha (i(a)\cdot (t,0,\dots , 0)).\) 
By \(f_\alpha \in \infty (\mathbb {G}, \mathbb {R})\)  and (REF ), \( \tilde{f}_\alpha \)  is \(\infty \)  map such that \(\frac{ \partial \tilde{f}_\alpha }{\partial p_1}(p) = X_1 f_\alpha (p) \ne 0\)  for all \(p\in S_\alpha \) . Hence we apply the classical Implicit Function Theorem to \(\tilde{f} _\alpha \)  and we conclude that also \(\hat{\phi }_\alpha \)  is \(\infty \) .
\(\mathbf {Step \, 4.}\)  Firstly we prove that \((\hat{\phi }_\alpha )_\alpha \) , up to subsequence, converges uniformly on \(\hat{ \mathcal {O}^{\prime }}\) , i.e. the condition (1) of the thesis follows.
In particular, we would like to show that there is a constant \(L>0\)  dependent on \( \Vert \nabla ^{\hat{\phi }} \hat{\phi }\Vert _{\mathcal {L}^\infty (\hat{ \mathcal {O}^{\prime }} , \mathbb {R}^{m-1})} \) , \(\mathcal {B}^{(1)}, \dots ,\mathcal {B}^{(n)}\)  such that for all \(\alpha >0\)  sufficiently small and all \(A\in \mathcal {O}^{\prime }\) , it holds
\(|\phi _\alpha (a)-\phi (a)|\le L\alpha .\) 
Therefore \((\phi _\alpha )_\alpha \) , up to subsequence, converges uniformly on \(\mathcal {O}^{\prime }\) . This is a direct consequence of Theorem REF . Here \(\nu ^\perp (p) =\mathbf {P}_\mathbb {W}(p)\) , \(\nu (p) =\mathbf {P}_p)\)  and \(E:=\mathcal {E}_{\hat{\phi }_\alpha } \)  is the intrinsic subgraph of \(\hat{\phi }_\alpha \) . Because \(\hat{\phi }_\alpha \)  is locally intrinsic Lipschitz, \(\mathcal {E}_{\hat{\phi }_\alpha } \)  is also a finite \(\mathbb {G}\) -perimeter set (see Theorem REF ).
For \(\beta \)  fixed as in Theorem REF  there is \(L=L(\beta , \mathcal {B}_M)>0\) , where \(\mathcal {B}_{M} = \max \lbrace  b_{ij}^{(s)} \, | \, i,j=1,\dots , m \, , s=1,\dots , n \rbrace \)  such that
\(\mathcal {U}((-L,0,\dots ,0),1) \subset \lbrace  q\, :\, \Vert  \mathbf {P}_\mathbb {W}q\Vert <-\beta \mathbf {P}_q̭ \rbrace  \quad \mbox{and} \quad \mathcal {U}((L,0,\dots ,0),1) \subset \lbrace  q\, :\, \Vert  \mathbf {P}_\mathbb {W}q\Vert <\beta \mathbf {P}_q̭ \rbrace \) 
Let \(p_a:= i(a) \cdot \hat{\phi }(i(a))\) , \(p^{\prime }_a:= i(a) \cdot (\phi (a)-\alpha L ,0,\dots , 0)\)  and \(p^{\prime \prime }_a:= i(a) \cdot (\phi (a)+\alpha L ,0,\dots , 0)\)  with \(a \in \mathcal {O}^{\prime } \)  and \(\alpha \in (0,1]\) . Notice that if we put \(r_0:= M+L+\max _{(x,y)\in \mathcal {O}^{\prime } }|x|_{\mathbb {R}^{m-1}}+ \max _{(x,y)\in \mathcal {O}^{\prime } }|y|_{\mathbb {R}^n}^{1/2} +\frac{1}{2} \mathcal {B}_M^{1/2} \max _{(x,y)\in \mathcal {O}^{\prime } }|x|_{\mathbb {R}^{m-1}}^{1/2} (M+L)^{1/2}< +\infty \)  then
\(\Vert  p^{\prime }_a\Vert  \le r_0,\quad \Vert  p^{\prime \prime }_a\Vert  \le r_0,\) 
for all \(a\in \mathcal {O}^{\prime }\) . Moreover by standard considerations (see [3]}) we know that for every \(q\in \mathcal {U}(0,r_0)\)  there exists \(c(r_0)>0\)  such that
\(\mathcal {U}(0,r)\cdot q \subset \mathcal {U}(q,c(r_0)\sqrt{r} ), \quad \forall \, r \in (0,1).\) 
So if we consider \(a\in \mathcal {O}^{\prime }\)  and \(\alpha \in (0,1]\) , we conclude that
\(\begin{aligned}\mathcal {U}(0,\alpha )\cdot p^{\prime }_a & \subset \mathcal {U}(p^{\prime }_a,c(r_0)\sqrt{\alpha }) \subset \lbrace  q\, :\, \Vert  \mathbf {P}_\mathbb {W}(p_a ^{-1} Q)\Vert <-\beta \Vert \mathbf {P}_p_a ^{-1} q) \Vert  \rbrace  \subset \mathcal {E}_{\hat{\phi }_\alpha }\\\mathcal {U}(0,\alpha )\cdot p^{\prime \prime }_a& \subset \mathcal {U}(p^{\prime \prime }_a,c(r_0)\sqrt{\alpha }) \subset \lbrace  q\, :\, \Vert  \mathbf {P}_\mathbb {W}(p_a ^{-1} Q)\Vert <\beta \Vert \mathbf {P}_p_a^{-1} q)\Vert  \rbrace  \subset \mathbb {G}- \mathcal {E}_{\hat{\phi }_\alpha }\end{aligned}\) 
More precisely, by definition of \(f_\alpha \)  we have
\(f_\alpha (p^{\prime }_a)=1, \quad \quad f_\alpha \left( i(a) \cdot \hat{\phi }_\alpha (i(a)) \right)=c, \quad \quad f_\alpha (p^{\prime \prime }_a)=0\) 
and so by \((\ref {90cit})\) , \((\ref {91cit})\)  and \((\ref {cit85})\)  we deduce
\(\phi (a)-\alpha L \le \phi _\alpha (a) \le \phi (a)+\alpha L\quad \mbox{ for all } \alpha \in (0,1], \, a\in \mathcal {O}^{\prime }.\) 
Hence (REF ) holds and consequently the condition (1) of the thesis is true.
\(\mathbf {Step \, 5.}\)  We prove that \((\hat{\phi }_\alpha )_{\alpha }\)  has a subsequence \((\hat{\phi }_k)_k\)  such that \(|\nabla ^{\hat{\phi }_k} \hat{\phi }_k |_{\mathbb {R}^{m-1}} \le \Vert  \nabla ^{\hat{\phi }} \hat{\phi }\Vert _{\mathcal {L}^\infty (\hat{ \mathcal {O}^{\prime }} ,\mathbb {R}^{m-1})}\)  for each \(k\in \mathbb {N}\)  on \(\mathcal {O} ^{\prime }\) , i.e. the condition \((2)\)  of the thesis holds.
Let \(\hat{\nabla }_\mathbb {G}f_\alpha :=(X_2f_\alpha , \dots ,X_mf_\alpha )\)  and \(\hat{\nu }_{\mathcal {E}} =(\nu ^{(2)}_{\mathcal {E}},\dots , \nu ^{(m)}_{\mathcal {E}})\) . Arguing as in Step 2 and by (), it follows
\(\begin{aligned}| \nabla ^{\hat{\phi }_\alpha } \hat{\phi }_\alpha (p)|_{\mathbb {R}^{m-1}} &= \frac{|\hat{\nabla }_\mathbb {G}f_\alpha (p)|_{\mathbb {R}^{m-1}}}{|X_1f_\alpha (p)|} \le \frac{1}{|X_1f_\alpha (p)|} \int _{ \mathcal {U}(0,\alpha )\cdot p} |\hat{\nu }_{\mathcal {E}} (q)| |\rho _\alpha (p\cdot q^{-1})| \, d |\partial \mathcal {E}|(q)\\& \le I_\alpha (p) \frac{ \Vert  \nabla ^{\hat{\phi }} \hat{\phi }\Vert _{\mathcal {L}^\infty (\hat{ \mathcal {O}^{\prime }},\mathbb {R}^{m-1})} }{|X_1f_\alpha (p)| \sqrt{1+ \Vert  \nabla ^{\hat{\phi }} \hat{\phi }\Vert ^2_{\mathcal {L}^\infty (\hat{ \mathcal {O}^{\prime }} ,\mathbb {R}^{m-1})} }} \\\end{aligned}\) 
and consequently by \((\ref {cit87})\)
\(| \nabla ^{\hat{\phi }_\alpha } \hat{\phi }_\alpha (p)|_{\mathbb {R}^{m-1}} \le \Vert  \nabla ^{\hat{\phi }} \hat{\phi }\Vert _{ \mathcal {L}^\infty (\hat{ \mathcal {O}^{\prime }} ,\mathbb {R}^{m-1})},\) 
for all \(p\in \hat{ \mathcal {O}^{\prime }}\) , as desired.
\(\mathbf {Step \, 6.}\)  Now we show that \((\hat{\phi }_\alpha )_\alpha \) , up to a subsequence, satisfying the condition \((3)\)  of the thesis, i.e. there exists a sequence \((\alpha _h)_h \subset (0,+\infty )\)  such that let \(\hat{\phi }_h \equiv \hat{\phi }_{\alpha _h}\)  we have
\(\nabla ^{\hat{\phi }_h} \hat{\phi }_h \rightarrow \nabla ^{\hat{\phi }} \hat{\phi }\quad \mbox{ a.e. in } \hat{ \mathcal {O}^{\prime }}.\) 
It is sufficient to show that there is a positive sequence \((\alpha _h)_h\)  converging to 0 such that there exists
\(\lim _{h\rightarrow \infty } \int _{ \mathcal {O}^{\prime }} \sqrt{ 1+|\nabla ^{\hat{\phi }_h} \hat{\phi }_h|_{\mathbb {R}^{m-1}} ^2} \, d \mathcal {L}^{m+n-1} = \int _{ \mathcal {O}^{\prime }} \sqrt{ 1+|\nabla ^{\hat{\phi }} \hat{\phi }|_{\mathbb {R}^{m-1}} ^2} \, d \mathcal {L}^{m+n-1}\) 
In fact, up a subsequence, using \((\ref {cit93})\)  and Proposition \(\ref {theo4.7}\)  we can assume that the sequence in \((\ref {citt98})\)  also satisfies
\(\nabla ^{\hat{\phi }_h} \hat{\phi }_h \rightarrow \nabla ^{\hat{\phi }} \hat{\phi }\quad \mbox{ weakly in }\, \mathcal {L}^1( \hat{ \mathcal {O}^{\prime }}, \mathbb {R}^{m-1} ).\) 
Consequently by Theorem  REF
\(\nabla ^{\hat{\phi }_h} \hat{\phi }_h \rightarrow \nabla ^{\hat{\phi }} \hat{\phi }\quad \mbox{ strongly in }\, \mathcal {L}^1( \hat{ \mathcal {O}^{\prime }}, \mathbb {R}^{m-1} )\) 
and so, up a subsequence, \((\ref {citt97})\)  holds.
In particular in order to show \((\ref {citt98})\)  we need to prove that there exist \(\hat{c} \in (0,1)\)  and a positive sequence \((\alpha _h)_h\)  converging to 0 satisfying
\(\exists \, \lim _{h\rightarrow \infty } |\partial \mathcal {E}_{\alpha _h, \hat{c} }|_\mathbb {G}( \mathcal {A}) = |\partial \mathcal {E}|_\mathbb {G}(\mathcal {A})\) 
where \(\mathcal {A}\)  is defined as in Step 2. Since the semicontinuity of \(\mathbb {G}\) -perimeter measure and Step 3 we obtain
\(|\partial \mathcal {E}|_\mathbb {G}(\mathcal {A} ) \le \liminf _{\alpha \rightarrow 0^+} |\partial \mathcal {E}_{\alpha , c }|_\mathbb {G}(\mathcal {A} )\) 
for every \(c\in (0,1)\) . More precisely by the last inequality and the coarea formula we get
\(\begin{aligned}|\partial \mathcal {E}|_\mathbb {G}(\mathcal {A}) & \le \int _0^1 \liminf _{\alpha \rightarrow 0^+} |\partial \mathcal {E}_{\alpha , c }|_\mathbb {G}(\mathcal {A} )\, dc \le \liminf _{\alpha \rightarrow 0^+} \int _0^1 |\partial \mathcal {E}_{\alpha , c }|_\mathbb {G}(\mathcal {A} )\, dc \\ & \quad = \liminf _{\alpha \rightarrow 0^+} \int _{\mathcal {A} } | \nabla _\mathbb {G}f_\alpha |\, d \mathcal {L}^{m+n} =: I(\hat{ \mathcal {O}^{\prime }} , c ).\end{aligned}\) 
Now using Step 2 we know that for every \(\hat{ \mathcal {O}}_0^{\prime }\)  such that \(\hat{ \mathcal {O}^{\prime }}\Subset \hat{ \mathcal {O}} _0^{\prime } \)  open and bounded there exists \((\alpha _h)_h \subset (0,+ \infty )\)  which converges to 0 and \( \bar{h} (\hat{ \mathcal {O}}_0^{\prime })>0\)  such that for all \(h\le \bar{h} (\hat{ \mathcal {O}}_0^{\prime })\)  we have
\(\int _{\mathcal {A} } | \nabla _\mathbb {G}f_{\alpha _h}| \, d \mathcal {L}^{m+n} \le |\partial \mathcal {E}|_\mathbb {G}( \mathcal {A}_0)\) 
where \(\mathcal {A}, \mathcal {A}_0\)  are defined as in Step 2. Consequently
\(I(\hat{ \mathcal {O}^{\prime }}, c ) \le |\partial \mathcal {E}|_\mathbb {G}(\mathcal {A}_0)\) 
for all \(c\in (0,1)\)  and \(\hat{ \mathcal {O}}_0^{\prime }\)  such that \(\hat{ \mathcal {O}^{\prime }} \Subset \hat{ \mathcal {O}}_0^{\prime } \)  open and bounded. Moreover by \(|\partial \mathcal {E}|_\mathbb {G}\)  is a Radon measure then by standard approximation argument we can rewrite \((\ref {cit109})\)  with \(\hat{ \mathcal {O}^{\prime }}\)  instead of \(\hat{ \mathcal {O}}_0^{\prime }\) . In particular using \((\ref {cit106})\)  we have that a.e. \(c\in (0,1)\)
\(\liminf _{\alpha \rightarrow 0^+} |\partial \mathcal {E}_{\alpha , c }|_\mathbb {G}(\mathcal {A} )= |\partial \mathcal {E}|_\mathbb {G}(\mathcal {A} ).\) 
Hence there exists \(\hat{c} \in (0,1)\)  and a positive sequence \((\alpha _h)_h\)  converging to 0 such that \((\ref {cit101})\)  follows.
Finally because \(\Vert  \phi \Vert _{\mathcal {L}^\infty (\mathcal {O} ^{\prime } , \mathbb {R}) } = M\)  we get
\(\partial \mathcal {E}\cap \mathcal {A} _1 = \partial \mathcal {E}\cap \mathcal {A} _2 = \emptyset \) 
where \( \mathcal {A}_1:=\lbrace p \in \hat{ \mathcal {O}^{\prime }} \cdot  :\, p_1 \le -2M \rbrace \)  and \(\mathcal {A}_2:=\lbrace p\in \hat{ \mathcal {O}^{\prime }} \cdot  :\, p_1 \ge 2M \rbrace \) .
Hence since Proposition \(\ref {citteo16OK}\)  and well-know \(\mathbb {G}\) -perimeter properties it follows that
\(\begin{aligned}\frac{1}{C(\mathbb {W},} \int _{ \mathcal {O}^{\prime }} \sqrt{1+| \nabla ^{\hat{\phi }} \hat{\phi }|_{\mathbb {R}^{m-1}} ^2} \, d\mathcal {L}^{m+n-1} & =|\partial \mathcal {E}|_\mathbb {G}( \hat{ \mathcal {O}^{\prime }} \cdot  \\& = |\partial \mathcal {E}|_\mathbb {G}(\mathcal {A} _1 )+ |\partial \mathcal {E}|_\mathbb {G}(\mathcal {A} )+ |\partial \mathcal {E}| ( \mathcal {A}_2 )\\& = |\partial \mathcal {E}|_\mathbb {G}( \partial \mathcal {E}\cap \mathcal {A}_1) + |\partial \mathcal {E}|_\mathbb {G}(\mathcal {A} ) + |\partial \mathcal {E}|_\mathbb {G}( \partial \mathcal {E}\cap \mathcal {A}_2 )\\&= |\partial \mathcal {E}|_\mathbb {G}( \mathcal {A} ).\end{aligned}\) 
In the same way, using also \((\ref {cit76})\) , \((\ref {cit79})\)  and \((\ref {90cit})\)
\(|\partial \mathcal {E}_{\alpha _h , c} |_\mathbb {G}( \mathcal {A} )= |\partial \mathcal {E}_{\hat{\phi }_h }|_\mathbb {G}( \mathcal {A} )\) 
and
\(|\partial \mathcal {E}_{\hat{\phi }_h }|_\mathbb {G}( \hat{ \mathcal {O}^{\prime }} \cdot = \frac{1}{C(\mathbb {W},} \int _{ \mathcal {O}^{\prime }} \sqrt{1+|\nabla ^{\hat{\phi }_h} \hat{\phi }_h |_{\mathbb {R}^{m-1}} ^2} \, \, d \mathcal {L}^{m+n-1}\) 
with \(\hat{\phi }_h = \hat{\phi }_{\alpha _h }\) . Now by the last three equalities and \((\ref {cit101})\) , \((\ref {citt98})\)  holds. Consequently \((\ref {citt97})\)  is true and the proof of theorem is complete.
